Physical Mathematics

Natalie Paquette’s remarkable essay explores the role of string theory in advancing problems in pure mathematics. The two fields of physics and mathematics have always been intimately mixed. Newton developed calculus in order to solve the problem of bodies moving under the influence of external forces and there is a long list of similar examples. Even when the two disciplines have been explored independently, it was often the case that mathematics, which had been created abstractly as an endeavor of pure thought, later found endless applications in the natural world. Group theory in crystallography, particle and condensed matter physics, Riemannian geometry in general relativity, and the matrix theory that formed the basis for Heisenberg’s approach to quantum mechanics are just a few well-known examples. The fact that mathematics was often ready in advance—without any connection having been made to the natural world in the minds of its founders—led Eugene Wigner in a celebrated essay to marvel at what he described as “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”1

Paquette’s essay is, in fact, concerned with the converse. The logic of string theory has led to new results in pure algebraic geometry that were not only unknown to mathematicians, but also came as a surprise to those outside physics. A complete understanding of Paquette’s essay requires advanced knowledge of string theory, and in particular of the work of Edward Witten, which I do not possess. Paquette has nonetheless carefully paved the way through topological field theories, the Seiberg-Witten approach to supersymmetric gauge theories, to the advanced subjects of Donaldson’s invariants, mirror symmetry, the remarkable relationship between a pair of distinct Calabi-Yau manifolds, and the link between the Monster group and conformal field theory.

Many physicists regard string theory, which does not currently allow for measurable verification and predictions, as some other form of mathematics. They would argue that what Paquette describes in her essay is merely mathematicians talking to other kinds of mathematicians. I think this view is mistaken. Before his tragic and untimely death last August, Joseph Polchinski published a remarked article, available on arXiv, entitled “Memories of a Theoretical Physicist.”2 Throughout his life, Polchinski, one of the most important physicists in the development of modern string theory, was motivated only by physical questions. For Polchinski, mathematics was just tools that he turned to and used as needed. I find myself in agreement with Paquette. We should expect to see more new insights in mathematics emerging from the rich structure of physical problems.